3 G Fractions algébriques 1 A. Définitions et vocabulaire : rappels (NAP P290 - AM P 253 partie A) Tous les nombres qui peuvent s’écrire sous la forme de fractions à termes entiers sont des nombres rationnels. L’ensemble des nombres rationnels est noté L’écriture est l’écriture d’une fraction où a est le Numérateur b est le Dénominateur différent de zéro a et b sont les TERMES de la fraction Une fraction algébrique ou rationnelle est une expression algébrique fractionnaire dont les termes sont des polynômes. Une fraction algébrique est aussi une division non exécutée, avec la différence que le diviseur et le dividende sont ici des expressions littérales. Exemples : , , et sont des fractions rationnelles n’est pas une fraction rationnelle B. Conditions d’existence (NAP P290 - AM P 253 partie B) 1) Notions Pour qu’une fraction existe c'est-à-dire désigne un réel, son dénominateur doit être différent de zéro. Cette condition est appelée condition d’existence de la fraction. 2) Méthode : Ecrire le polynôme « dénominateur » et de ne pas l’égaler à zéro ; Résoudre l’équation ainsi formée ; Ecrire les conditions d’existence en excluant les valeurs trouvées dans la résolution de l’équation. 3) Exemples : • La fraction existe c'est-à-dire ne représente un nombre réel que (si et seulement si) son dénominateur n’est pas nul x – 3 0 x 3 x \ 3 La condition d’existence de est x 3 Nous noterons CE pour désigner la condition d’existence. b a 3 x 4 4x x 2 - + - 2 x 5 x - + 2 x 5 x - + 9 a 5 2 - x 5 x + 3 x 4 4x x 2 - + - Û Û ¹ Û ¹ Â Î { } 3 x 4 4x x 2 - + - ¹ La version d’essai
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3 G Fractions algébriques 1
Chapitre VII : Fractions algébriques
A. Définitions et vocabulaire : rappels (NAP P290 - AM P 253 partie A) Tous les nombres qui peuvent s’écrire sous la forme de fractions à termes entiers sont des
nombres rationnels.
L’ensemble des nombres rationnels est noté
L’écriture est l’écriture d’une fraction où a est le Numérateur
b est le Dénominateur différent de zéro
a et b sont les TERMES de la fraction
Une fraction algébrique ou rationnelle est
une expression algébrique fractionnaire dont les termes sont des polynômes.
Une fraction algébrique est aussi une division non exécutée, avec la différence que le diviseur et le
dividende sont ici des expressions littérales.
Exemples : , , et sont des fractions rationnelles
n’est pas une fraction rationnelle
B. Conditions d’existence (NAP P290 - AM P 253 partie B) 1) Notions
Pour qu’une fraction existe c'est-à-dire désigne un réel,
son dénominateur doit être différent de zéro.
Cette condition est appelée condition d’existence de la fraction.
2) Méthode :
Ecrire le polynôme « dénominateur » et de ne pas l’égaler à zéro ;
Résoudre l’équation ainsi formée ;
Ecrire les conditions d’existence en excluant les valeurs trouvées dans la résolution de l’équation. 3) Exemples :
• La fraction existe c'est-à-dire ne représente un nombre réel que
(si et seulement si) son dénominateur n’est pas nul x – 3 0 x 3 x \ 3
La condition d’existence de est x 3
Nous noterons CE pour désigner la condition d’existence.
ba
3x44xx2
-+-
2x5x
-+
2x5x
-
+
9a52 -
x5x +
3x44xx2
-+-
ÛÛ ¹Û ¹ ÂÎ { }
3x44xx2
-+-
¹
La version d’essai
3 G Fractions algébriques 2
• La fraction existe si t 3 et t -3
En effet 0 (t – 3) (t + 3) 0
(t – 3) 0 et (t + 3) 0 t 3 et t -3 t \ -3 ; 3
Les conditions d’existence de sont t 3 et t -3
4) Exercices : Détermine les conditions d’existence (CE) de chacune des fractions suivantes
Série 1.
9t4
2 -¹ ¹
9t2 - ¹Û ¹Û ¹ ¹Û ¹ ¹ ÂÎ { }
9t4
2 -¹ ¹
2a1-
4t4
2 -
3b3a3b3a
-+
3)1)(x(x6
--
65xx4x
2 ++-
23
2
xx5x
--
La version d’essai
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3 G Fractions algébriques 3
Série 2. NAM P 125 n°C (AM P131 n°C)
53+xx
132-
-cc
725 +a
)4(4+-bbb
22
233 --
-
xx
x
9
152 -
+
a
a
xx +
-27
yy
y
82
13
2
-
-
Conditions existence d’une fraction: exercices
La version d’essai
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3 G Fractions algébriques 4
12
42 +- xx
x
623
123 +++
+-
xxx
x
ab5
cba-3
ba +-1
225
yx -
baa36
2-
Conditions existence d’une fraction: exercices
La version d’essai
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3 G Fractions algébriques 5
C. Simplification
1) Notions (NAM P290-291 - AM P 253 c )
On peut multiplier ou diviser les deux termes d’une fraction par un même nombre non nul,
on obtient une fraction équivalente ( égale). avec a et b, c , m
= = = =
Simplifier une fraction rationnelle,
c’est diviser les deux termes de cette fraction par un même polynôme non nul.
2) Méthode pour simplifier une fraction algébrique (NAM P 290 - AM P 253 )
Pour factoriser une fraction rationnelle, il faut :
Factoriser les polynômes des deux termes de la fraction ;
Énoncer les conditions d’existence (CE) de cette fraction ;
Diviser les deux termes de la fraction par un même polynôme non nul ;
Enoncer les conditions de simplification (CS)
3) Conditions de simplification (NAM P 291 - AM P 254 )
Les valeurs qui annulent le polynôme par lequel on divise les deux termes de la fraction rationnelle s’appellent les conditions de simplification (notées CS).
La condition d’existence d’une fraction n’est pas toujours identique à celle de sa forme simplifiée
Exemple :
= =
La condition de simplification précise que l’égalité n’est vraie que pour x 3.
4) Remarque
Après factorisation du numérateur et/ou du dénominateur, il arrive parfois qu'un facteur du numérateur soit l'opposé d'un facteur du dénominateur. Voici comment les rendre égaux.
Exemple : = = = =
º ÎÂ Î 0Â
m . bm . a
ba
m . cm . b m . a +
m . cb) (a . m +
cb a +
.b 2a . 2
ba
xx
x
3292
-
-)3()3)(3(
-+-
xxxx
xx 3+
¹
94
462 -
-
x
x)32)(32(
)23(2-+
-xxx
)32)(32()23(2+-+
--xxx
)23)(32()23(2xx
x-+
--)32(
2+
-x
CS : x – 3 0 x 3
D’abord FACTORISER avant de simplifier.
CE : x2- 3x 0 x (x – 3) 0
x 0 et (x – 3) 0 x 0 et x 3
0 x 3
CE : x 0
INFO
INFO. (-1)
. (-1)
La version d’essai
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3 G Fractions algébriques 6
5) Exercices
Série 1. Factoriser ou ne pas factoriser ? Simplifier ou ne pas simplifier ? Coche (...x...) les exercices pour lesquels tu ne peux factoriser ni le numérateur, ni le dénominateur. Pour les exercices non cochés, factorise, puis simplifie si cela est possible. Sinon, recopie la fraction telle quelle.
Simplification de fractions- CE : exercices
La version d’essai
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3 G Fractions algébriques 7
Simplification de fractions- CE : exercices
La version d’essai
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Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
3 G Fractions algébriques 8
Série 2. Associe à chaque fraction sa forme simplifiée. (AM P 134 )
= o o
= o o
= o o
= o o
= o o
= o o
22
2
--x
xx13
2-xx
264-xx
3yx +
yxyx33
22
+-
3)( yx +-
622 yx +
yx2
xyyx33
22
--
3yx -
2
2
8
4
xy
yx2x
Simplification de fractions- CE : exercices
La version d’essai
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
3 G Fractions algébriques 9
Série 3. Simplifie les fractions (NAM P 126 C - AM P 132 série c )
3
2
5
3
ab
ba
34
5
x
x-
32
54
16
12
ba
ba-
3
23
15
25
xyz
zxy
ba
ab3
3
)(3)(5baba
++
)(15)(12 2
yxxyxx
++-
Simplification de fractions- CE : exercices
La version d’essai
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
3 G Fractions algébriques 10
)(5)(3abba
--
)2)(3()2)((bayxbayx---+
)(5))(32(
xyxyxxy
---
4
632 -
-
x
x
12462
++
aa
Simplification de fractions- CE : exercices
La version d’essai
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
3 G Fractions algébriques 11
22 ba
bxax
-
-
9
962
2
-
+-
a
aa
abba5544
--
xx4433 2
+-
xx
525252
--
2
2
26
3
bab
aba
-
-
Simplification de fractions- CE : exercices
La version d’essai
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
3 G Fractions algébriques 12
aba
baa
33
)(42
222
+
-
xx
x
2
632 -
+
65
121232
2
++
++
xx
xx
23
222
23
++
--+
xx
xxx
Simplification de fractions- CE : exercices
La version d’essai
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
Cat
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3 G Fractions algébriques 13
D. Opérations sur les fractions rationnelles
1) Réduction au même dénominateur (AM P 254 )
a) Méthode : pour réduire des fractions rationnelles au même dénominateur, il faut :
b) Exemples
et et et
et et et
et et et
c) Remarque: Ce n’est généralement pas une bonne idée d’effectuer la multiplication des binômes au dénominateur. La plupart du temps, il est plus avantageux de laisser le dénominateur sous la forme d’un produit, parce qu’ainsi, on peut plus facilement effectuer d’autres simplifications par la suite.
d) Exercices Recherche du dénominateur commun PPCM PPCM PPCM
a et b a2 et a5 et
2a et 3a x et x3 et
xy et yz 3 a2 et 9a et
10 x et 15 y 6 a4 et 15 a3 et
Factorise le dénominateur. Détermine le dénominateur commun et réduits les fractions au même D
et
et
29b
256b
522 ba
a
- 5b5aa- 2a
3- 44aa
5 - a2 +-
32
3
2b .9b
b .2 2
3 .6b
3 . 55 b)b)(a(a
a+- b)5(a
a- 2)(a
3- 22)(a
5) - (a
-
5
3
b 18
b 4
b 18
155 b)b)(a(a . 5
a 5.+- b)b)(a5(a
b)(a a+-
+22)(a
2) - (a . 3
- 22)(a
5) - (a
-
2x
2
x
5
3x
3a2x
2b
3 x 2x+ 5 x
3x+
x2
3x 5+
4
22 -x
x42
3+xx
22 b2aba
a
+- bab-
Factoriser les polynômes des deux termes de la fraction (et indiquer les CE.)
Simplifier, si possible, les facteurs communs d’une même fraction rationnelle.
Déterminer le dénominateur commun qui est le PPCM des dénominateurs (il s’obtient en
multipliant tous les facteurs communs ou non, chacun d’eux affecté de son plus grand
exposant)
Multiplier le numérateur et le dénominateur de chaque fraction par les facteurs qui
permettent d’obtenir le dénominateur commun.
INFO
La version d’essai
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3 G Fractions algébriques 14
2) Addition et soustraction de fractions rationnelles
a) Méthode : pour additionner ou soustraire des fractions algébriques, il faut :
Remarque :
Ce n’est généralement pas une bonne idée d’effectuer la multiplication des binômes au dénominateur. La plupart du temps, il est plus avantageux de laisser le dénominateur sous la forme d’un produit, parce qu’ainsi, on peut plus facilement effectuer d’autres simplifications par la suite